3. Selforganization in Mathematics

As early as 1952, the English mathematician Alan Turing published a paper entitled “ The chemical basis of morphogenesis” in which he suggests that:


"a system of chemical substances, called morphogens, reacting together and diffusing through a tissue is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances."



Here Turing already stated the main principles of selforganization. Furthermore, he suggests that reaction-diffusion processes play an important role in morphogenesis. He continues to illustrate his point with the aid of a mathematical model, a set of differential equations the solution of which gives the development of the concentration pattern of morphogens over time. This mathematical approach has been proven to be very valuable, because it provides a level of abstraction on which different dynamic systems can be compared with each other. For instance, Troy (1978) showed that the dynamics of the Belousov-Zhabotinski reaction and the dynamics of an action potential according to the model of Hodgkin and Huxley are mathematically closely related. More important is however, that where thermodynamics cannot tell us what happens after a dynamic system has become unstable (Part 2), mathematics can! Differential equations describe the behaviour of a dynamic system in a deterministic way, without referring to the microscopic fluctuations, which were shown to be so important for the onset of selforganization. Then how can these deterministic equations provide a meaningful model for selforganization? The answer is simple: analytical methods to solve the equations can no longer be applied and numerical methods are used. The computational inaccuracies implicit in these methods play the role of fluctuations, perturbing the system from an unstable state and leading it away. Since Turing, a lot of theoretical models have been developed explaining pattern formation during development by processes of selforganization (Reviews: Brenner et al., 1981; Meinhardt, 1978). Experimental evidence for the involvement of selforganization in development is still confined to the aggregation of slime molds (Robertson and Cohen, 1972; Tomchik and Devreotes, 1981).


Up till now we have mostly dealt with structures in 'real space'. It is important to realize that selforganizing processes can also explain the formation of 'time structures', of which a periodic oscillation is a simple example. An extensive review of "the geometry of biological time" is given by Winfree (1980). A dynamic system, governed by simple (nonlinear) kinetics, is able to display very complicated behaviour: jump transitions between several steady states, simple and compound oscillations, non-periodic oscillations ('chaos'), bursts (Rossler and Wegman. 1977).


An important point is, that this complex behaviour is not a property of a small class of exotic dynamic systems: nearly all nonlinear dynamic systems will have interesting (chaotic, periodic) regions in their 'parameter­ space' (Helleman,1980). There are two important classes of dynamic systems: Hamiltonian systems in which there are no terms equivalent to 'friction' and motion is pertained indefinitely without external drive (the harmonic oscillator is an example) and dissipative systems which must be continuously driven to prevent motion to die out. Only dissipative systems are relevant for our discussion. Dissipative systems differ from Hamiltonian ones, in that they possess 'attractors': modes of behaviour to which neighbouring solutions are attracted. This property is a direct consequence of dissipation and it ensures stability of a certain type of behaviour: any perturbation away from it, will eventually die out. Behaviour on an attractor can be simple (equilibrium or steady state; point attractor), periodic (limit cycle or toroidal attractor) or chaotic ('strange attractor'). Although the behaviour of a system on such a strange attractor may look quite erratic, it represents a highly ordered structure: the dynamic variables which describe the behaviour of the system are not allowed to take on any possible value, but instead the behaviour of the system is limited to a specific volume in phase space (the space spanned by all the dynamic variables). Inside this volume resides the strange attractor, which is a fractal (cf. Mandelbrot,1983 for an extensive discussion of fractals): a scale-invariant structure, consisting of a single trajectory (the path a system follows in phase space) which never closes on itself, filling the restricted phase volume inhomogeneously (cf. Abraham and Shaw, 1983 for a 'visual' introduction to chaos and strange attractors).


The behaviour of dissipative dynamic systems can be switched between the different (stationary, periodic, chaotic) modes by changing the value of a nonlinearity parameter (usually a 'friction' parameter regulating the level of dissipation). When the value of such a parameter is gradually increased, a series of 'bifurcations' is observed. In a bifurcation one at tractor becomes unstable, while at the same time another attractor (possibly of a completely different type) appears. As a result, the system changes its behaviour abruptly. The following scenario is often found. First the system is stable: any perturbation from this steady state will die away. Increasing the parameter gradually changes the state, until at a critical value of the parameter the steady state becomes unstable. At this point the system 'bifurcates': it changes its behaviour abruptly to an oscillatory regime, which is again stable against perturbations. Further increase of the parameter leads to a succession of bifurcations in which the complexity of the oscillations increases. These 'period doublings' follow each other ever faster, until at another critical value of the parameter, a chaotic regime is entered. Here the behaviour of the system can hardly be distinguished from a stochastic one.


Depending on the type of system this chaotic behaviour can be more or less oscillatory in nature (reviews: Schuster, 1984; Holden, 1986; Olsen and Degn, 1986). This route to chaos via a converging sequence of period doublings has some very universal characteristics (Cvitanovic, 1983). There are at least two alternative routes (Eckman, 1981): a short one with chaos after only three bifurcations (Ruelle and Takens, 1971) and a route via 'intermittent' behaviour (Pommeau and Manneville, 1980; Manneville and Pommeau, 1980). When the parameter is further increased in the chaotic region, a number of 'windows' may appear where the behaviour is again periodic. Clearly, the system can be made to behave (pseudo-)stochastically or deterministically, simply by changing an external parameter value.


In summary, most nonlinear dynamic systems are capable of producing a wealth of 'time-structures' including chaotic ones. Transitions among different types of behaviour appear very suddenly and are governed by the value of one or more parameters relating to the level of dissipation. When spatial dependence and diffusion are taken into account, the spontaneous formation of ordered spatial structures out of an initially homogeneous state is possible. Both temporal and spatial aspects of selforganization seem to be relevant to brain function and development.